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## Neighborhood Grey-Level Difference Matrix (NGLDM)

The neighborhood grey-level difference matrix (NGLDM) corresponds to the difference of grey-level between one voxel and its 26 neighbours in 3 dimensions (8 in 2D). Three texture indices can be computed from this matrix. An element $$(i,1)$$ of NGLDM corresponds to the probability of occurrence of level $$i$$ and an element $$(i,2)$$ is equal to:
\begin{equation}
NGLDM(i,2)= \sum_{p}\sum_{q} \left\lbrace
\begin{array}{ll}
|\overline{M}(p,q)-i| \mbox{~if $I(p,q)=i$} \\
0 \mbox{~else}
\end{array}
\right.
\end{equation}

where $$\overline{M}(p,q)$$ is the average of intensities over the 26 neighbour voxels of voxel $$(p,q)$$.

NGLDM_Coarseness is the level of spatial rate of change in intensity.

\begin{equation}
NGLDM\_Coarseness=\frac{1}{\sum_{i} NGLDM(i,1) \cdot NGLDM(i,2)}
\end{equation}

NGLDM_Contrast is the intensity difference between neighbouring regions.

\begin{equation}
NGLDM\_Contrast=\left[ \sum_{i} \sum_{j} NGLDM(i,1) \cdot NGLDM(j,1) \cdot (i-j)^{2} \right] \cdot \frac{\sum_{i} NGLDM(i,2)}{E \cdot G \cdot (G-1)}
\end{equation}

where E corresponds to the number of voxels in the Volume of Interest and G the number of grey-levels.

NGLDM_Busyness is the spatial frequency of changes in intensity.
\begin{equation}
\begin{split}
NGLDM\_Busyness=\frac{\sum_{i} NGLDM(i,1) \cdot NGLDM(i,2)}{\sum_{i} \sum_{j} \left | i \cdot NGLDM(i,1) - j \cdot NGLDM(j,1) \right | } \\
with~ NGLDM(i,1)\neq 0,~ NGLDM(j,1)\neq 0
\end{split}
\end{equation}